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Generalization of Stoney's equation for flexoelectric thin films on elastic substrates

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Pith's one-line read This paper derives explicit closed-form generalizations of Stoney's equation that predict curvature and stretching strain of a piezoelectric-flexoelectric thin film on an elastic substrate, opening curvature metrology to electromechanical f

desk verdict A genuinely new Stoney extension for flexoelectric films, but the closed-circuit formulas rest on a questionable swap between two energy forms that differ by a boundary term. read the letter →

arxiv 2601.09160 v1 pith:GBZGZ6VT submitted 2026-01-14 physics.app-ph

Swarnava Ghosh This is my paper · ORCID
classification physics.app-ph
keywords flexoelectricitypiezoelectricityStoney'sequationthinfilmssubstratecurvatureelectromechanicalcouplingstraingradientfilm-substratebilayer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper generalizes Stoney's classical relation between film stress and substrate curvature to thin films that are both piezoelectric and flexoelectric. It derives closed-form formulas for the midplane stretching strain and curvature of a film-substrate bilayer, for uniform and linearly graded film properties, in both open-circuit (direct flexoelectric) and closed-circuit (applied-voltage, converse) configurations. If correct, these formulas let experimenters extract electromechanical film constants from curvature measurements in the same way residual stress is measured today, and they predict how applied voltage bends the bilayer. A notable conclusion is that the electric polarization in the film varies linearly through the thickness, and that Stoney's original estimates increasingly overestimate stretching strain and curvature as the film becomes thicker and stiffer relative to the substrate.

What carries the argument

The carrying mechanism is a total enthalpy functional for the film–substrate bilayer: elastic, dielectric, piezoelectric, and flexoelectric energy densities integrated over volume, with flexoelectricity represented in two forms—a direct form coupling strain gradient to electric field (voltage from deformation) and a converse form coupling strain to electric-field gradient (deformation from voltage). The argument proceeds by requiring this functional to be stationary with respect to the midplane stretching strain ε0 and the curvature κ, which yields two coupled algebraic equations whose solution is the closed-form generalization. In the non-uniform case, the same stationarity machinery is app

What would settle it

A finite-element solution of the full coupled equilibrium equations using one flexocoupling tensor and explicit surface terms—rather than the paper's two separate enthalpy densities—computed for the same geometry and constants as Figures 3 to 5 would settle the matter: if its predicted stretching strain and curvature do not match Eqs. (51)–(54) in both open and closed circuits, the two-enthalpy treatment is not equivalent.

Watch

Extended reading notes

Core claim

The paper's central claim is that electromechanical coupling in a thin film can be incorporated into explicit, energy-minimized expressions for the bilayer's midplane stretching strain ε0 and curvature κ. For uniform film properties it reports closed-form formulas (Eqs. 51–54) that connect ε0 and κ to the film's piezoelectric constant e31, flexocoupling constant µ12, permittivity k33, elastic moduli, film and substrate thicknesses, mismatch strain, and applied voltage, with separate branches for closed-circuit (converse) and open-circuit (direct) conditions; a linearly varying mismatch strain leads to the analogous formulas Eqs. (94)–(97). The paper shows these expressions reduce to the pure

Load-bearing premise

The load-bearing premise is that direct and converse flexoelectricity can be modeled by two distinct enthalpy densities that differ by an integration-by-parts boundary term; if a single flexoelectric tensor together with the correct boundary conditions changes the open- and closed-circuit curvature predictions, the paper's central contrast between the two configurations would not survive.

Editorial extensions

If this is right

  • Curvature metrology can be extended from residual stress to electromechanical properties: measuring κ and ε0 in open- and closed-circuit configurations gives access to e31 and µ12 as well as the mismatch strain.
  • An applied voltage bends the bilayer, with the bending direction and magnitude set by the film/substrate thickness and stiffness ratios—so the formulas double as a design rule for flexoelectric actuators on compliant substrates.
  • In the closed-circuit/converse case with no piezoelectricity, flexoelectricity alone leaves the curvature and stretching strain unchanged; the open-circuit/direct case does change them, so comparing the two configurations separates flexoelectric from piezoelectric effects.
  • For films with a linear mismatch gradient, the derived expressions let a single curvature/strain measurement be decomposed into average and gradient contributions to the film stress.
  • The polarization is linear through the film thickness and tied to both ε0 and κ, giving an independent surface measurement that can corroborate the curvature-derived constants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The difference between open- and closed-circuit curvature of the same sample could be used to identify e31 and µ12 separately without knowing the residual stress, because voltage and boundary conditions enter the formulas through different terms—a protocol the paper motivates but does not spell out.
  • Because the direct and converse flexoelectric enthalpy densities differ only by an integration-by-parts boundary term, the predicted open/closed contrast should be cross-checked against a formulation with one flexoelectric tensor plus consistent surface terms before taking the quantitative difference at face value.
  • The same energy-minimization framework could accommodate finite deformations, anisotropic or inhomogeneous substrates, and non-axisymmetric buckling, with the present formulas serving as the linear benchmark.
  • The linear-through-thickness polarization prediction could be tested directly on ultrathin films via surface-potential or second-harmonic measurements, which would also test the constitutive assumption that flexoelectricity couples to strain gradient rather than to strain itself.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper derives closed-form extensions of Stoney's equation for a film/substrate bilayer in which the film is piezoelectric and flexoelectric. Assuming Kirchhoff kinematics, transverse isotropy, and axial symmetry, it solves the one-dimensional electrostatic field for open- and closed-circuit conditions, integrates the enthalpy over the thickness, and minimizes with respect to stretching strain and curvature. Explicit formulas are given for uniform properties (Eqs. 51–54) and for a linear mismatch-strain gradient (Eqs. 94–97), followed by parametric plots and polarization expressions. The stated goal is to enable curvature-based extraction of film electromechanical properties.

Significance. The linear-elastic, axisymmetric framework is natural, and the paper is explicit about its variational principle, uses no fitted parameters, and provides closed-form expressions. If the closed-circuit branch is correct, the work is a practical extension of Stoney metrology to piezo/flexo films. However, the central open/closed contrast is tied to an unproven choice of enthalpy density; the most striking result—that flexoelectricity disappears in a closed circuit for e31=0, Eq. (57)—is not supported by a boundary-consistent calculation. The direct/open-circuit part and the elastic/piezoelectric reductions are plausible, but the closed-circuit formulas need revision.

major comments (2)
  1. [§2 and §4, Eqs. (5)–(6), (49), (57)] The two flexoelectric enthalpy densities are related by ψ_conv − ψ_dir = (µ_ijlk ε_ij E_l),k, a pure divergence. The paper adopts ψ_conv for the closed circuit and ψ_dir for the open circuit, but minimizes the volume integral (49) without including the corresponding surface term. This is not invariant after eliminating the electrostatic field and minimizing over κ, because κ changes ε at the electrode surfaces. A concrete failure is Case IV: for e31=0 the closed-circuit solution (32) gives E_z=V/h_f, so ψ_conv=0 and Eq. (57) follows, whereas the paper's own direct form (5) gives −2µ12 ε_rr,z E_z = 2µ12 κ V/h_f ≠ 0 in the same state. Thus the disappearance of the flexoelectric effect in the closed circuit, and more generally the open/closed contrast in §4.1, is not established unless the missing surface work is included or shown to vanish. Please re-derive with a single energy and consist
  2. [§4, Eqs. (12)–(13), (40), (51)–(52)] The statement that the stress and electric displacement expressions hold irrespective of the choice of enthalpy density is true for the bulk constitutive laws, but it does not justify replacing the volume enthalpy in Eq. (49). Because the two forms differ by a boundary term, the reduced plate energy is changed by a κ-dependent surface contribution. This affects the closed-circuit formulas: Eq. (40) is proportional to e31, so it vanishes when e31→0 and leads to Eq. (57); using the direct form (5) instead gives a leading closed-circuit flexoelectric term proportional to µ12 V κ/h_f. The open/closed contrast is therefore a consequence of the chosen energy gauge rather than of the physics, unless the authors can prove that the boundary term cancels identically for the stated boundary conditions.
minor comments (5)
  1. [Eq. (37)] The dielectric enthalpy density for the direct case is missing a factor: −(1/2)k33(E^Dir_z)^2 should contain the coefficient 2e31/k33, not e31. The integrated result in Eq. (46) appears consistent with the corrected density, but the displayed formula is misleading.
  2. [Eqs. (61), (73)] Eq. (61) uses k instead of κ for the curvature in the strain expression; Eq. (73) writes the substrate elastic enthalpy with M^f instead of M^s. These typos should be fixed.
  3. [§3, after Eq. (27)] The statement that the electric displacement in the substrate is zero because piezoelectricity and flexoelectricity are absent is not correct as written: for a dielectric substrate, D = kE. Either the substrate is intended to be conductive or the electric field is assumed to be screened; this assumption should be stated explicitly.
  4. [§4, Eq. (33) and Case IV] The open-circuit electric field (33) is derived for e31 ≠ 0 and contains factors of 1/e31. The e31 → 0 limit used in Case IV is well-defined for the integrated quantities, but this should be stated explicitly to avoid confusion about division by zero.
  5. [§4.1 and §5.1, Eqs. (51)–(54), (94)–(97)] The final formulas are very long and no intermediate algebra or symbolic verification is provided. Given the typos in nearby equations, the authors should supply a supplementary derivation or a machine-checked symbolic verification of these expressions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the generalized Stoney formulas follow from a stated variational principle with externally tabulated material parameters; no fitted input is renamed as a prediction.

full rationale

The derivation is self-contained. Starting from the enthalpy densities (Eqs. 2-6) and the variational principle (Eq. 9), the paper solves the equilibrium equations (Eqs. 28-30) for the electric field in the open/closed cases (Eqs. 32-33), integrates the enthalpy density (Eq. 49), and obtains strain and curvature from the stationarity conditions (Eq. 50). The resulting closed-form expressions (Eqs. 51-54 and 94-97) are algebraic outputs of these steps. No parameter is fitted to the curvatures or strains that are then 'predicted': the material constants in Table 1 are taken from prior independent literature, and the elastic-only limits (Eqs. 55-56, 98-99) reproduce Freund's cited results. The author's previous DFT papers are cited only as background context and are not load-bearing for the Stoney generalization. The only substantive concern is physical, not circular: the direct and converse flexoelectric enthalpy forms (Eqs. 5-6) differ by a boundary term, so treating them as separate models for open and closed circuits may make the predicted open/closed difference (e.g., Eq. 57) depend on a gauge choice. That is a modeling/boundary-condition question about assumptions, not a case where an output is identical to an input by construction. No circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard continuum electromechanics plus the paper-specific modeling choices: an electrically inert substrate and two flexoelectric enthalpy forms. No new physical entities are introduced; no numbers are fitted to data in this paper.

assumptions (5)
  • domain assumption Enthalpy decomposition and constitutive laws of linear electroelasticity including flexoelectricity (Eqs. 1-6), from [56].
    The derivation relies on this standard continuum model for piezoelectric and flexoelectric dielectrics.
  • domain assumption Kirchhoff thin-plate kinematics for the substrate: σzz=0, straight normals, axially symmetric deformation with uniform curvature and stretching strain (Section 3).
    This is the standard modeling assumption for thin film/substrate curvature problems, following Freund.
  • ad hoc to paper Film is transversely isotropic, poled along z, and isotropically flexoelectric; substrate has no electrical coupling and is assumed to have zero electric displacement (E=0 in substrate).
    The substrate's electrically inert nature is never explicitly justified (e.g., as a conductor), yet it is load-bearing for the closed-circuit case.
  • ad hoc to paper Two distinct flexoelectric enthalpy densities (Eqs. 5 and 6) are used respectively for open- and closed-circuit configurations; these forms differ by a boundary term and may not represent independent physics.
    This modeling choice determines the difference between the direct and converse results; it is not justified from first principles.
  • domain assumption For the non-uniform example, the mismatch strain is linear in z with constant material properties (Eq. 88).
    This is a specific illustrative case chosen for tractability.

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Pith. "Pith review of Generalization of Stoney's equation for flexoelectric thin films on elastic substrates." pith.science (2026). https://pith.science/paper/GBZGZ6VT

@misc{pith2026260109160,
  author       = {Pith},
  title        = {Pith review of: Generalization of Stoney's equation for flexoelectric thin films on elastic substrates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBZGZ6VT}},
  note         = {Machine review of arXiv:2601.09160}
}
read the original abstract

When a thin film is deposited on an incompatible elastic substrate, the film develops an elastic mismatch strain, causing the film-substrate system to bend. Stoney's equation relates the curvature of the bent film-substrate system with the residual stress developed in the film, and can be used to infer film properties from curvature measurements. Certain materials exhibit electromechanical coupling, such as piezoelectricity and flexoelectricity, which can alter the curvature and strains. In this work, we generalize Stoney's equation to include flexoelectric and piezoelectric effects in the film. Considering both open and closed circuit configurations, as well as uniform and non-uniform film properties, we compare different cases of electromechanical coupling and discuss their influence on curvature, strains, and electric polarization in the film.

Figures

Figures reproduced from arXiv: 2601.09160 by the authors.

Figure 1
Figure 1. Top figure shows a film deposited on a flexible substrate. The film has an elastic mismatch and is held in a flat configuration by an applied stress σm. Upon removal of this applied stress, the film-substrate system deforms as shown in the bottom. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The film substrate system for (a) converse case with applied voltage V across the height of the film and (b) direct case without any applied voltage is shown. We consider a film-substrate bilayer system where the elastic mismatch strain εm and the elastic moduli of the film and the substrate are constant. We analyze two cases. In the first case, a voltage of V is applied across the film layer. This is called a close… view at source ↗
Figure 3
Figure 3. Normalized stretching strains and curvatures of the film-substrate system for various ratios of film￾substrate thickness hf /hs and stiffness Mf /Ms. (a) show the stretching strain and (d) the curvature when only elastic effects are considered (case II). (b), (c) show the stretching strains and (e), (f) the curvatures for converse and direct effects when both piezoelectric and flexoelectric effects are present (case… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Normalized stretching strains and curvatures for the converse case shown as a function of the ratio of the thickness of film and substrate for different cases of stiffness ratios Mf /Ms. Figures (a)-(c) show the stretching strains, and (d)-(f) show the curvatures. diff…
Figure 5
Figure 5. Figure 5: Normalized stretching strains and curvatures for the direct case shown as a function of the ratio of the thickness of film and substrate for different cases of stiffness ratios Mf /Ms. Figures (a)-(c) shows the stretching strains, and (d)-(f) shows the curvatures. We a…
Figure 6
Figure 6. Figure 6: Contour plot showing the effect of potential on the normalized stretching strains in (a) - (b) and normalized curvatures in (c) - (d) for the converse case shown for different ratios of the thickness of film and substrate. ( e31(z) + e15(z) ) ∂ 2ϕ ∂r∂z − 1 r ( µ11(z) −…
Figure 7
Figure 7. Figure 7: A film substrate system with non-uniform film property and non-uniform elastic mismatch strain. A non-uniform stress is applied to the film to maintain the flat configuration. Upon removing the stress, the system deforms to a bent configuration. For closed circuit conf…
Figure 8
Figure 8. Figure 8: Normalized stretching strains and curvatures of the film-substrate system for various ratios of film￾substrate thickness hf /hs and stiffness Mf /Ms with non-uniformly varying elastic mismatch strain with constant gradient and constant material properties. (a) depict t…
Figure 9
Figure 9. Figure 9: Normalized stretching strains and curvatures for the converse case shown as a function of the ratio of the thickness of film and substrate, shown for different cases of stiffness ratios Mf /Ms with nonuniform elastic mismatch strain. Figures (a)-(c) show the stretching…
Figure 10
Figure 10. Figure 10: Normalized stretching strains and curvatures for the direct case shown as a function of the ratio of the thickness of film and substrate, shown for different cases of stiffness ratios Mf /Ms with nonuniform elastic mismatch strain. Figures (a)-(c) show the stretching …

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.